Duality and equilibrium prices in economics of uncertainty

A random variable (RV) X is given a minimum selling price (S) S-U(X) := sup(x)(x + EU(X - x)) and a maximum buying price (B) B-p(X) := inf(x + EP(X - x)) where U(.) and P(.) are appropriate functions. These prices are derived from considerations of stochastic optimization with recourse, and are called recourse certainty equivalents (RCE's) of X. Both RCE's compute the "value" of a RV as an optimization problem, and both problems (S) and (B) have meaningful dual problems, stated in terms of the Csiszar phi-divergence I-phi (p,q) := (i=1)Sigma(n) q(i) phi (pi/qi) a generalized entropy function, measuring the distance between RV's with probability vectors p and q. The RCE S-U was studied elsewhere, and applied to production, investment and insurance problems. Here we study the RCE Bp, and apply it to problems of inventory control (where the attitude towards risk determines the stock levels and order sizes) and optimal insurance coverage, a problem stated as a game between the insurance company (setting the premiums) and the buyer of insurance, maximizing the RCE of his coverage.